Physics-Informed Fine-Tuning for physics discovery from random and sparse data

Although significant advances have been made in both numerical methods and machine learning approaches for solving differential equations across various scientific sectors, these methods often rely on complete information about the differential equations, including precise parameter values, which are not easily obtainable in real-world scenarios. To address this challenge, data-driven methods for discovering differential equations have gained growing popularity in recent years. However, many existing approaches demand unrealistic prerequisites, such as extensive high-fidelity data or carefully designed low-fidelity data and functions. In this paper, we propose a novel method: P hysics- I nformed F ine- T uning ( PIFT ) to discover the unknown parameters in differential equations when only randomly distributed sparse data points are available. PIFT consists of three stages; (i) generating low fidelity data from prior knowledge under realistic settings, (ii) pre-training a single neural network with the generated low fidelity data, and (iii) fine-tuning the pre-trained model using physics-informed loss function. PIFT is evaluated on seven scientific problems including five ordinary differential equations and two partial differential equations. We also demonstrate the robustness and generalizability of PIFT to out-of-distribution tasks. PIFT exhibits high accuracy and robustness in discovering unknown parameters of differential equations from randomly distributed sparse data points.